Chapter 8Chapter 8

Rotational Equilibrium Rotational Equilibrium

andand

Rotational DynamicsRotational Dynamics

TorqueTorque

Torque, , is the tendency of a Torque, , is the tendency of a force to rotate an object about force to rotate an object about some axissome axis

is the torqueis the torque– symbol is the Greek tausymbol is the Greek tau

F is the forceF is the force d is the d is the lever armlever arm (or moment arm) (or moment arm)

Fd

Direction of TorqueDirection of Torque

Torque is a vector quantityTorque is a vector quantity The direction is perpendicular to the The direction is perpendicular to the

plane determined by the lever arm and plane determined by the lever arm and the forcethe force

For two dimensional problems, into or out For two dimensional problems, into or out of the plane of the paper will be sufficientof the plane of the paper will be sufficient

If the turning tendency of the force is If the turning tendency of the force is counterclockwise, the torque will be counterclockwise, the torque will be positivepositive

If the turning tendency is clockwise, the If the turning tendency is clockwise, the torque will be negativetorque will be negative

Units of TorqueUnits of Torque

SISI Newton x meter = NmNewton x meter = Nm

US CustomaryUS Customary foot x pound = ft lbfoot x pound = ft lb

Lever ArmLever Arm The lever arm, d, is The lever arm, d, is

the the perpendicularperpendicular distance from the distance from the axis of rotation to a axis of rotation to a line drawn from the line drawn from the axis of rotation to a axis of rotation to a line drawn along line drawn along the the direction of the the direction of the forcethe force d = L sin d = L sin ΦΦ

An Alternative Look at An Alternative Look at TorqueTorque

The force could The force could also be resolved also be resolved into its x- and y-into its x- and y-componentscomponents The x-component, The x-component,

F cos F cos Φ, Φ, produces 0 produces 0 torquetorque

The y-component, The y-component, F sin F sin Φ, produces Φ, produces a non-zero torquea non-zero torque

Torque, finalTorque, final

From the components of the force From the components of the force or from the lever arm,or from the lever arm,

F is the forceF is the force L is the distance along the objectL is the distance along the object Φ is the angle between the force and Φ is the angle between the force and

the objectthe object

sinFL

Net TorqueNet Torque

The net torque is the sum of all the The net torque is the sum of all the torques produced by all the forcestorques produced by all the forces Remember to account for the Remember to account for the

direction of the tendency for rotationdirection of the tendency for rotation Counterclockwise torques are positiveCounterclockwise torques are positive Clockwise torques are negativeClockwise torques are negative

Torque and EquilibriumTorque and Equilibrium

First Condition of EquilibriumFirst Condition of Equilibrium The net external force must be zeroThe net external force must be zero

This is a necessary, but not sufficient, This is a necessary, but not sufficient, condition to ensure that an object is in condition to ensure that an object is in complete mechanical equilibriumcomplete mechanical equilibrium

This is a statement of translational This is a statement of translational equilibriumequilibrium

0Fand0F

0F

yx

Torque and Equilibrium, Torque and Equilibrium, contcont

To ensure mechanical equilibrium, To ensure mechanical equilibrium, you need to ensure rotational you need to ensure rotational equilibrium as well as translationalequilibrium as well as translational

The Second Condition of The Second Condition of Equilibrium statesEquilibrium states The net external torque must be zeroThe net external torque must be zero

0

Mechanical EquilibriumMechanical Equilibrium

In this case, the First In this case, the First Condition of Condition of Equilibrium is Equilibrium is satisfiedsatisfied

The Second The Second Condition is not Condition is not satisfiedsatisfied Both forces would Both forces would

produce clockwise produce clockwise rotationsrotations

N500N5000F

0Nm500

Axis of RotationAxis of Rotation

If the object is in equilibrium, it does not If the object is in equilibrium, it does not matter where you put the axis of matter where you put the axis of rotation for calculating the net torquerotation for calculating the net torque The location of the axis of rotation is The location of the axis of rotation is

completely arbitrarycompletely arbitrary Often the nature of the problem will suggest Often the nature of the problem will suggest

a convenient location for the axisa convenient location for the axis When solving a problem, you When solving a problem, you mustmust specify specify

an axis of rotationan axis of rotation Once you have chosen an axis, you must maintain Once you have chosen an axis, you must maintain

that choice consistently throughout the problemthat choice consistently throughout the problem

Center of GravityCenter of Gravity

The force of gravity acting on an The force of gravity acting on an object must be consideredobject must be considered

In finding the torque produced by In finding the torque produced by the force of gravity, all of the the force of gravity, all of the weight of the object can be weight of the object can be considered to be concentrated at considered to be concentrated at one pointone point

Calculating the Center of Calculating the Center of GravityGravity

The object is The object is divided up into a divided up into a large number of large number of very small very small particles of weight particles of weight (mg)(mg)

Each particle will Each particle will have a set of have a set of coordinates coordinates indicating its indicating its location (x,y)location (x,y)

Calculating the Center of Calculating the Center of Gravity, cont.Gravity, cont.

The torque The torque produced by each produced by each particle about the particle about the axis of rotation is axis of rotation is equal to its equal to its weight times its weight times its lever armlever arm

Calculating the Center of Calculating the Center of Gravity, cont.Gravity, cont.

We wish to locate the point of We wish to locate the point of application of the application of the single forcesingle force , , whose magnitude is equal to the whose magnitude is equal to the weight of the object, and whose weight of the object, and whose effect on the rotation is the same effect on the rotation is the same as all the individual particles.as all the individual particles.

This point is called the This point is called the center of center of gravitygravity of the object of the object

Coordinates of the Center Coordinates of the Center of Gravityof Gravity

The coordinates of the center of The coordinates of the center of gravity can be found from the sum gravity can be found from the sum of the torques acting on the of the torques acting on the individual particles being set equal individual particles being set equal to the torque produced by the to the torque produced by the weight of the objectweight of the object

i

iicg

i

iicg m

ymyand

m

xmx

Center of Gravity of a Center of Gravity of a Uniform ObjectUniform Object

The center of gravity of a The center of gravity of a homogenous, symmetric body homogenous, symmetric body must lie on the axis of symmetry.must lie on the axis of symmetry.

Often, the center of gravity of such Often, the center of gravity of such an object is the an object is the geometricgeometric center center of the object.of the object.

Experimentally Experimentally Determining the Center of Determining the Center of GravityGravity

The wrench is hung The wrench is hung freely from two different freely from two different pivotspivots

The intersection of the The intersection of the lines indicates the lines indicates the center of gravitycenter of gravity

A rigid object can be A rigid object can be balanced by a single balanced by a single force equal in force equal in magnitude to its weight magnitude to its weight as long as the force is as long as the force is acting upward through acting upward through the object’s center of the object’s center of gravitygravity

Notes About EquilibriumNotes About Equilibrium

A zero net torque does not mean A zero net torque does not mean the absence of rotational motionthe absence of rotational motion An object that rotates at uniform An object that rotates at uniform

angular velocity can be under the angular velocity can be under the influence of a zero net torqueinfluence of a zero net torque

This is analogous to the translational This is analogous to the translational situation where a zero net force does not situation where a zero net force does not mean the object is not in motionmean the object is not in motion

Solving Equilibrium Solving Equilibrium ProblemsProblems

Draw a diagram of the systemDraw a diagram of the system Isolate the object being analyzed Isolate the object being analyzed

and draw a free body diagram and draw a free body diagram showing all the external forces showing all the external forces acting on the objectacting on the object For systems containing more than For systems containing more than

one object, draw a separate free body one object, draw a separate free body diagram for each objectdiagram for each object

Problem Solving, cont.Problem Solving, cont.

Establish convenient coordinate Establish convenient coordinate axes for each object. Apply the axes for each object. Apply the First Condition of EquilibriumFirst Condition of Equilibrium

Choose a convenient rotational Choose a convenient rotational axis for calculating the net torque axis for calculating the net torque on the object. Apply the Second on the object. Apply the Second Condition of EquilibriumCondition of Equilibrium

Solve the resulting simultaneous Solve the resulting simultaneous equations for all of the unknownsequations for all of the unknowns

Example of aExample of aFree Body DiagramFree Body Diagram

Isolate the object Isolate the object to be analyzedto be analyzed

Draw the free Draw the free body diagram for body diagram for that objectthat object Include all the Include all the

external forces external forces acting on the acting on the objectobject

Example of aExample of aFree Body DiagramFree Body Diagram

The free body The free body diagram includes diagram includes the directions of the directions of the forcesthe forces

The weights act The weights act through the through the centers of gravity centers of gravity of their objectsof their objects

Fig 8.12, p.228

Slide 17

Torque and Angular Torque and Angular AccelerationAcceleration

When a rigid object is subject to a When a rigid object is subject to a net torque (net torque (≠0), it undergoes an ≠0), it undergoes an angular accelerationangular acceleration

The angular acceleration is directly The angular acceleration is directly proportional to the net torqueproportional to the net torque The relationship is analogous to The relationship is analogous to ∑F = ∑F =

mama Newton’s Second LawNewton’s Second Law

Moment of InertiaMoment of Inertia

The angular acceleration is The angular acceleration is inversely proportional to the inversely proportional to the analogy of the mass in a rotating analogy of the mass in a rotating systemsystem

This mass analog is called the This mass analog is called the moment of inertia, moment of inertia, I, of the objectI, of the object

SI units are kg mSI units are kg m22

2mrI

Newton’s Second Law for Newton’s Second Law for a Rotating Objecta Rotating Object

The angular acceleration is directly The angular acceleration is directly proportional to the net torqueproportional to the net torque

The angular acceleration is The angular acceleration is inversely proportional to the inversely proportional to the moment of inertia of the objectmoment of inertia of the object

I

More About Moment of More About Moment of InertiaInertia

There is a major difference between There is a major difference between moment of inertia and mass: the moment of inertia and mass: the moment of inertia depends on the moment of inertia depends on the quantity of matter quantity of matter and its and its distributiondistribution in the rigid object. in the rigid object.

The moment of inertia also depends The moment of inertia also depends upon the location of the axis of upon the location of the axis of rotationrotation

Moment of Inertia of a Moment of Inertia of a Uniform RingUniform Ring

Image the hoop is Image the hoop is divided into a divided into a number of small number of small segments, msegments, m11 … …

These segments These segments are equidistant are equidistant from the axisfrom the axis

22ii MRrmI

Other Moments of InertiaOther Moments of Inertia

Rotational Kinetic EnergyRotational Kinetic Energy

An object rotating about some axis An object rotating about some axis with an angular speed, with an angular speed, ω, has ω, has rotational kinetic energy ½Iωrotational kinetic energy ½Iω22

Energy concepts can be useful for Energy concepts can be useful for simplifying the analysis of simplifying the analysis of rotational motionrotational motion

Total Energy of a SystemTotal Energy of a System

Conservation of Mechanical EnergyConservation of Mechanical Energy

Remember, this is for conservative Remember, this is for conservative forces, no dissipative forces such as forces, no dissipative forces such as friction can be presentfriction can be present

fgrtigrt )PEKEKE()PEKEKE(

Angular MomentumAngular Momentum

Similarly to the relationship between Similarly to the relationship between force and momentum in a linear force and momentum in a linear system, we can show the relationship system, we can show the relationship between torque and angular between torque and angular momentummomentum

Angular momentum is defined as Angular momentum is defined as L = I L = I ωω

and and t

L

Angular Momentum, contAngular Momentum, cont

If the net torque is zero, the angular If the net torque is zero, the angular momentum remains constantmomentum remains constant

Conservation of Linear MomentumConservation of Linear Momentum states: The angular momentum of a states: The angular momentum of a system is conserved when the net system is conserved when the net external torque acting on the external torque acting on the systems is zero.systems is zero. That is, when That is, when

ffiifi IIorLL,0

Problem Solving HintsProblem Solving Hints

The same basic techniques that The same basic techniques that were used in linear motion can be were used in linear motion can be applied to rotational motion.applied to rotational motion. Analogies: F becomes , m becomes I Analogies: F becomes , m becomes I

and a becomes , v becomes and a becomes , v becomes ω and ω and x becomes θx becomes θ

More Problem Solving More Problem Solving HintsHints

Techniques for conservation of energy Techniques for conservation of energy are the same as for linear systems, as are the same as for linear systems, as long as you include the rotational long as you include the rotational kinetic energykinetic energy

Problems involving angular momentum Problems involving angular momentum are essentially the same technique as are essentially the same technique as those with linear momentumthose with linear momentum The moment of inertia may change, leading The moment of inertia may change, leading

to a change in angular momentumto a change in angular momentum